Compressed scan systems

ABSTRACT

A method for building a fast scan system is provided in which a scanner moves the scan sensors faster than scanners of the prior art, even though the total distance that the scan sensors move longer. The scan system includes (a) a scan sensor that measures the scan target by moving around it, and (b) a data processing system that calculates a parameter of the scan target from the collected data. The scan sensor, which has a limited sensing bandwidth, is moved along multiple paths along the target at a scan speed that is faster than the scan speed determined by the scan sensor bandwidth, so as to obtain a clear signal directly from the scan sensor output. The target is then recovered from the scan output using a compressed sampling data recovery data processing method.

CROSS REFERENCE TO RELATED AFFLICTIONS

The present application is related to, and claims priority of,provisional patent application Ser. No. 61/169,923, entitled “CompressedSampling for Scan Systems with Limited Sensing Bandwidth”, M-17596-V1US, filed on Apr. 16, 2009 The provisional patent application is herebyincorporated by reference in its entirety.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to scan systems, such as those usingscanning electron microscope (SEM) or atomic force microscopy (AFM). Inparticular, the present invention relates to how to build a fast scansystem.

2. Discussion of the Related Art

In a conventional scan system (e.g., SEM, scanning tunneling microscope(STM), AFM, magnetic force microscopy (MFM), single-dish radiotelescopes, scanning radar system, or a scanning laser Doppler system),a scanner scans the target line-by-line. Often the scan time is so longthat it requires a lot of resources (e.g., money, energy) to run such ascanner. In some other case, since the scan time is long, it isdifficult to scan a dynamically changing target.

SUMMARY

The present invention provides a method for building a fast scan system.Under this method, a scanner moves the scan sensors faster than scannersof the prior art, even though the total distance that the scan sensorsmove longer.

According to one embodiment of the present invention, a scan systemincludes (a) a scan sensor that measures the scan target by movingaround it, and (b) a data processing system that calculates a parameterof the scan target from the collected data.

According to one embodiment of the present invention, the scan sensor,which has a limited sensing bandwidth, is moved along multiple pathsalong the target. The scan speed is faster than the scan speeddetermined by the scan sensor bandwidth, so as to obtain a clear signaldirectly from the scan sensor output. The target is then recovered fromthe scan output using a compressed sampling data recovery dataprocessing method.

The present invention is better understood upon consideration of thedetailed description below in conjunction with the accompanyingdrawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an example of a periodic torus scan path with X=1,Y=1, n_(x)=4, n_(y)=5, x₀=:1, and y₀=0.

FIG. 2 illustrates an example of a Ping-pong scan path, when X=1, Y=1,n_(x)=4, n_(y)=5, x₀=:1, and y₀=0.

FIG. 3 illustrates an example of a projection from spatial 2DFTcomponents of a target to 1DFT components of the output of a scan sensorfor a Torus scan.

FIG. 4 illustrates another example of a projection from spatial 2DFTcomponents of a target to 1DFT components of the output of a scan sensorfor a Torus scan.

FIG. 5 illustrates the relationship between 2DCT of the target and 1DFTof the output of a scan sensor when the scan sensor is only sensitive atlow frequencies for a ping-pong scan.

FIG. 6 illustrates an example for selecting suitable scan numbers.

FIG. 7 illustrates an example 2DCT component that can be measured by aband pass scan sensor for a ping-pang scan.

FIG. 8 illustrates another example 2DCT component that can be measuredby a band-pass scan sensor for a ping-pang scan.

FIG. 9 illustrates an example of a Lissajous scan path.

FIG. 10 illustrates an example of a Lissajous-like scan path.

FIG. 11 illustrates an example of a daisy scan path.

FIG. 12 illustrates a frequency selection of a linearly selecting scanpath.

FIG. 13 illustrates an example of a ping-pong scan with atwo-dimensional scan sensor array. There are four sensors in the figure,forming a 2-by-2 array, each sensor having a different offset and adifferent scan area.

FIG. 14 illustrates an example of a ping-pong scan with a linear scansensor array. There are 3 sensors in the figure, forming a 1-by-3 array,each sensor having a different offset and a different scan area.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Scan System

According to one embodiment of the present invention, scan systemincludes (a) scan sensor that measures the scan target by moving aroundit and (b) a data processing system that calculates the scan target fromthe collected data.

The present application is applicable to a scanning electron microscope(SEM), scanning tunneling microscope (STM), atomic force microscope(AFM), magnetic force microscopy (MFM), single-dish radio telescopes,scanning radar system, scanning laser Doppler system, and so on.

Scan Target

A scan system measures a physical parameter of a scan target. Forexample, in a SEM system, the target can be a surface morphology of acomputer chip. For another example, in a radar system, the target can beobjects in an area of the sky. The target can be 2-dimensional or3-dimensional.

Scan Sensor

A scan sensor is the device in a scan system that measures a physicalvalue of the target at or around a “scan sensing location” on thetarget. The system may move the scan sensing location on the target andthus measure a spatial property of the target. The path traversed by asensing location on the target, as it is being moved about, is called ascan path. In a preferred embodiment, the scan sensor's output and thescan sensing location are each measured as a function of time and issent to a data processing system to calculate the physical parameter ofthe target. Many scan sensors, their sensing locations, and methods forcollection of scan sensor output data and sensing location data areknown in existing scan systems, such as SEM, STM, AFM, MFM, single dishtelescopes and scanning radar systems. In these systems, a scan sensormay take different forms and may be moved in different ways. Forexample, in an AFM system, a sensor may be a small cantilever with asharp tip at the end, which measures an atomic force of the target whileit is moved by a piezo actuators in, for example, orthogonal (“x” and“y”) directions. The sensor is often most sensitive when the scan outputis around the natural frequency of the cantilever.

As another example, in a SEM system, an electron beams is directed atthe target. The electron beam is moved by deflection coils to scan atarget area. An electron detector or an x-ray detector measures theelectron or x-ray scattered from the target when the electron beam is atdirected at different locations.

As another example, in a laser Doppler system, a laser beams is directedat the target. A mirror that rotates in two directions can direct thelaser beam in many different directions. A laser Doppler sensor measuresthe velocity of the target when the laser beam is pointed to differentdirections.

Sensors of a scan system often have limited bandwidth, which oftenlimits their available scan speeds. For example, the cantilevers used inan AFM system are only sensitive around the resident frequency, theelectron detector used in an SEM might be sensitive only at lowfrequencies; the bolometer used in radio telescope only has limitedresponse time; the analog-to-digital converters in different kinds ofscan systems only have limited sampling frequency.

Scan Method

In many traditional scan systems, a scan sensor has to be moved slowlyenough to avoid blurring the scan signal. Particularly, the scan speedshould be slow enough that the bandwidth of the scan output is less thanthe bandwidth of the sensor. Denoting the measurement result (the finaloutput of the whole scan system) of the scan system as T_(m)(x; y), thescan sensor's sensitive location (x(t); y(t)) may be moved as a functionof time. In a traditional scan system, the bandwidth of the apparentscan output, defined as g_(sm)(t)=T_(m)(x(t); y(t)) is less than thebandwidth of the sensor.

According to one embodiment of the present invention, the scan sensormay be moved at a faster speed than the traditional scan system. Whilethe scan output may be blurred because of the higher speed, the targetimage may be recovered from the blurred signal by a data processingsystem. As result, the apparent scan output can have a bandwidth that iswider than the sensor bandwidth.

Scan Paths

The scan path is designed both to provide an effective cover of thetarget, and to easily operable by the system.

Multiple Paths

According to one preferred embodiment, the scan sensor is moved alongmultiple scan paths. The scan paths are designed so that one or moredifferent aspects of the target are measured by a different path.

Continuous Scan Path

In a preferred embodiment, the scan paths are continuous lines. Thebenefit is that the scan output is continuous and hence its does nothave high frequency components that are caused by sudden jumps in thesignal. Another benefit is that the scan sensor is kept on the targetcontinuously because, in some systems, moving a scan sensor out of thetarget is difficult or slow.

Closed Scan Path

In a preferred embodiment, the scan paths are closed (i.e., the endingpoint results from steering the scan path back to the ending point). Oneadvantage is that if the scan sensor moves along the scan linerepetitively, its output is periodic and continuous. It is preferred toscan the target along the scan path an integer number of times because,so doing, the Fourier transform of the scan output only has values thatare integer multiples of the scan frequency. Here, scan frequency equals1 over the time required for a sensor to be moved along the scan pathonce. One benefit is that the frequency aliasing problem, which happensfor non-repetitive signals, can be avoided. Another benefit is that ifthe signal-to-noise ratio of the scan sensor output is not high, thescan path may be repeated multiple times and noise is reduced by theaveraging process.

Closed Scan and Spatial Transform

In one preferred embodiment, the scan path, or the set of scan paths,are designed so that there is each frequency band of the scan sensoroutput are contributed, or mainly contributed, by a portion of thespatial frequency components of the target. For different scan paths,different portion of the spatial frequency components contribute todifferent frequency bands of the sensor output. These scan paths arecalled selecting scan paths. The advantage of a selecting scan path isthat it enables selecting a desired spatial transform components by theuse of different scan paths. For example, one method makes the spatialcomponents measured by a set of scan paths to be well-distributed in thespatial transform space. As another example, one method may avoidmeasuring same components repetitively by the use of different scanpaths.

One set of selecting scan paths are referred to as linear selecting scanpaths. In such scan paths, a linear relationship exists between thespatial frequency components and frequency components of the scan sensoroutput. FIG. 12 illustrates a frequency selection of a linearlyselecting scan path.

Another set of selecting scan path are referred to as projecting scanpaths. In a projecting scan path, each frequency component of the scansensor output is contributed by only a finite number of spatialfrequency components of the target. One advantage of a projecting scanpath is that it facilitates calculating the expected scan sensor outputfrom the target or calculating the spatial transform components of thetarget from the scan sensor output. This enables a fast target recoveryalgorithm in a data processing system.

A different scan path is associated with different spatial transforms.The present invention provides a class of methods for designing scanpaths and their relationship with different spatial transforms.

The Torus Scan Path

In a preferred embodiment, the scan paths are provided as torus paths.The benefit is that there is a simple relationship between the 2DFT ofthe target and 1DFT of the scan output along the scan path. For atwo-dimensional square target, the scan path of a torus scan can bedescribed by:

x(t)=v _(x) t+x ₀ mod 2X;

y(t)=v _(y) t+y ₀ mod 2Y.

where t denotes time, v_(x) denotes the velocity in x direction andv_(y) denotes the velocity in y direction. The target is within therange from −X to X and from −Y to Y.To make the path closed,

${v_{x} = {2\; X\; n_{x}\frac{1}{T}}};$$v_{y} = {2\; Y\; n_{y}{\frac{1}{T}.}}$

where T is the scan period. n_(x) and n_(y) are integers that calledscan numbers. n_(x) and n_(y) are relatively prime to each other.

The 2DFT component at frequency (f_(x), f_(y)) with amplitude G(f_(x);f_(y)) contributes to the scan sensor output:

${G\left( {f_{x};f_{y}} \right)}^{\; 2{\frac{1}{4}{\lbrack{{{({{f_{x}n_{x}} + {F_{y}n_{y}}})}\frac{t}{T}} + \frac{f_{x}x_{0}}{2\; X} + \frac{f_{y}y_{0}}{2\; Y}}\rbrack}}}$

which has frequency at

$f_{s} = {\frac{1}{T}{\left( {{f_{x}n_{x}} + {f_{y}n_{y}}} \right).}}$

This equation is referred to as the scan frequency equation.

The scan frequency equation shows that the 2DFT components of the targetare linearly projected to the 1DFT components of the scan sensor output.If the scan sensor is has a limited bandwidth, the scan sensor outputcomes from a ‘strip’ of 2DFT components of the target in the spatialfrequency domain. Only the frequency components within the strip can bemeasured by the band limited sensor.

If a different scan path with a different (n_(x); n_(y)) is used, theresult is a rotation of the measured strip in the spatial frequencydomain of the target. Hence, we can select a set of strips in thespatial frequency domain to measure by selecting different (n_(x);n_(y)) according to the scan frequency equation.

Denoting integer n_(s)=f_(s)T as the scan frequency number,

n _(s) =f _(x) n _(x) +f _(y) n _(y).

This equation is referred to as the scan frequency number equation,which is an integer equation, as all its variables are integers. Hence,a 2DFT frequency component at (f_(x), f_(Y)) only contributes to 1DFTcomponent at scan frequency number n_(s). However, it is possible thatmultiple 2DFT components contribute to the same 1DFT component of thescan output. In fact, the frequency number equation may be used in manydifferent scan paths. Achieving (n_(x); n_(y)) such that only one 2DFTcomponent contributes to each 1DFT component of the scan sensor output,and solving the frequency number equation in general, are discussed inthe section in the following which introduces the scan number equation.

The relationship between the 2DFT components of the target and the 1DFTcomponents of the scan sensor output enables a fast calculation of theexpected scan output from the target using the fast Fourier transform.One can simply calculate the 1DFT component from 2DFT components bysolving the scan number equation:

FIG. 1 illustrates an example of a periodic torus scan path with X=1,Y=1, (n_(x); n_(y)) n_(x)=4, n_(y)=5, x₀=:1, and y₀=0.

FIG. 3 illustrates an example of a projection from spatial 2DFTcomponents of a target to 1DFT components of the output of a scan sensorfor a Torus scan. FIG. 4 illustrates another example of a projectionfrom spatial 2DFT components of a target to 1DFT components of theoutput of a scan sensor for a Torus scan. FIG. 6 illustrates an exampleof selecting suitable scan numbers.

The Ping-Pong Scan

FIG. 13 illustrates an example of a ping-pong scan with atwo-dimensional scan sensor array. There are four sensors in the figure,forming a 2-by-2 array, each sensor having a different offset and adifferent scan area. FIG. 14 illustrates an example of a ping-pong scanwith a linear scan sensor array. The ping-pong scan is a wrapped linearscan, having the wrapping function:

${R\left( {u;U} \right)} =^{\text{?}}\begin{matrix}{{u\; {mod}\; U};} & {{if}\mspace{14mu} b\frac{u}{\bigcup}c} & {{even};} \\{{U_{i}\left( {u\; {mod}\; U} \right)};} & {{if}\mspace{14mu} b\frac{u}{\bigcup}c} & {{odd}.}\end{matrix}$ ?indicates text missing or illegible when filed

where U is the wrapping boundary.

The path of a constant speed ping-pong scan is given by

x(t)=R(v _(x) t+x ₀ ;X);

y(t)=R(v _(y) t+y ₀ ;Y).

Note the target for a ping-pang scan is defined on x2[0; X] and y2[0;Y]. The ping-pong scan is both periodic and continuous.

Denoting the two dimensional cosine transform (2DCT) of the target as

${C\left( {f_{x};f_{y}} \right)} = {\text{?}{g\left( {x;y} \right)}{\cos \left( {{1/4}\frac{f_{x}x}{X}} \right)}{\cos \left( {{1/4}\frac{f_{y}y}{Y}} \right)}{dxdy}}$?indicates text missing or illegible when filed

the 2DCT component at (f_(x); f_(y)) contributes to 1DFT component ofthe scan sensor output at frequency

$f_{s} = {\frac{1}{T}{\left( {{f_{x}n_{x}} + {f_{y}n_{y}}} \right).}}$

This is the same scan frequency equation as the scan frequency equationdiscussed in the torus scan section above. The scan number equation fora ping-pong scan is also the same scan number equation for the torusscan:

$f_{s} = {\frac{1}{T}{\left( {{f_{x}n_{x}} + {f_{y}n_{y}}} \right).}}$

The 1DFT component with scan frequency number n_(s) isg(n_(s))=′n_(s)=f_(x)n_(x)+f_(y)n_(y) 4C(f_(x); c_(y)). Thus, g(n_(s))is the sum of all the 2DCT components that have the samen_(s)=f_(x)n_(x)+f_(y)n_(y). Achieve (n_(x); n_(y)) such that only one2DCT component contributes to one 1DFT component of the scan sensoroutput, and solving the frequency number equation in general, arediscussed in the section which introduces the scan number equation.

FIG. 2 illustrates an example of a Ping-pong scan path, when X=1, Y=1,n_(x)=4, n_(y)=5, x₀=:1, and y₀=0. FIG. 5 illustrates the relationshipbetween 2DCT of the target and 1DFT of the output of a scan sensor whenthe scan sensor is only sensitive at low frequencies for a ping-pongscan. FIG. 7 illustrates an example 2DCT component that can be measuredby a band pass scan sensor for a ping-pang scan. FIG. 8 illustratesanother example 2DCT component that can be measured by a band-pass scansensor for a ping-pang scan.

A Scan with a Continuous VelocityIn some embodiments, it is much easier to implement scan paths with acontinuous scan velocity.

The Lissajous Scan

In on embodiment, the scan paths can be designed as a Lissajous scan,which is given by:

${{x_{L}(t)} = {\cos \left( {{1/4}\frac{\left( {{v_{x}t} + x_{0}} \right)}{X}} \right)}};$${y_{L}(t)} = {{\cos \left( {{1/4}\frac{\left( {{v_{y}t} + y_{0}} \right)}{Y}} \right)}.}$

A Lissajous scan can be made periodic by:

v_(x)=2Xvn_(x);

v_(y)=2Yvn_(y).

Such a Lissajous scan is periodic with a continuous path and acontinuous velocity.The spatial transform associated with a Lissajous scan is theTwo-dimensional Chebyshev polynomial transform (2DCPT) of the2-dimensional spatial function f(x; y).

${®\left( {c_{x};c_{y}} \right)} = {\frac{4}{1/4^{2}}\text{?}\text{?}{T_{c_{x}}(x)}{T_{c_{y}}(y)}{f\left( {x;y} \right)}p\frac{dxdy}{\left( {1_{i}x^{2}} \right)\left( {1_{i}y^{2}} \right)}\text{:}}$?indicates text missing or illegible when filed

where T_(n) is Chebyshev polynomials of the First Kind. T_(n) is definedas:

T _(n)(x)=cos(narc cos x)

f(x; y) can be calculated from its 2DCPT as

The scan frequency equation for a Lissajous scan is given by:

$f_{s} = {\frac{1}{T}\left( {{c_{x}n_{x}} + {c_{y}n_{y}}} \right)}$

The scan number equation for a Lissajous scan is given by:

n _(s) =c _(x) n _(x) +c _(y) n _(y).

Achieving (n_(x); n_(y)) such that only one 2DCPT component contributesto one 1DFT component of the scan sensor output, and solving thefrequency number equation in general, are discussed in the sectionintroducing the scan number equation.

FIG. 9 illustrates an example of a Lissajous scan path.

The General Lissajous scan

In some embodiments, other scan paths may be provided to meet thespecific needs of a system. For example, one may construct a targetfunction K_(n)(x) as:

K _(n)=cos(nt)

x=h(R(t;1))

R(u; U) is the wrapping function which is discussed above in conjunctionwith the ping-pong scan.

${R\left( {u;U} \right)} =^{\text{?}}{\begin{matrix}{{u\; {mod}\; U};} & {{if}\mspace{14mu} b\frac{u}{\bigcup}c} & {{even};} \\{{U_{i}\left( {u\; {mod}\; U} \right)};} & {{if}\mspace{14mu} b\frac{u}{\bigcup}c} & {{odd}.}\end{matrix}\text{?}}$ ?indicates text missing or illegible when filed

where h(t), referred to as a template function, is a function defined ont2[0; 1]. h(t) is preferably continuous and increasing. Denoting thederivative of h(t) with respect to t as h⁰(t), K_(n) forms a sequence oforthogonal polynomials with respect to the weight function

?${\text{?}{K_{n}(x)}{K_{m}(x)}\frac{dx}{h^{0}(t)}} = {\text{?}\begin{matrix}0 & {\text{:}n\text{?}} \\\frac{1}{4} & {{\text{:}n} = {m = 0}} \\{\frac{1}{4} = 2} & {{:n} = {m = 0}}\end{matrix}}$ ?indicates text missing or illegible when filed

Using the target function, a scan path may be provided as

x=h _(x)(t)

y=h _(y)(t)

where h_(x)(t) and h_(y) (t) are template functions in x and y directionrespectively.The target function in the x direction is denoted as K_(c) _(x) (x), andthe target function in the y direction is denoted as J_(cy)(y).

K _(cx)(t)=cos(c _(x) t)

x(t)=h _(x)(t)

J _(c) _(y) (t)=cos(c _(y) t)

x(t)=h _(y)(t)

For a two-dimensional image, the 2-dimensional K transform (2DKT) isgiven by as:

${{f\left( {x;y} \right)} = {\text{?}\text{?}^{-}\left( {c_{x};c_{y}} \right){K_{c_{x}}(x)}{J_{c_{y}}(y)}}};\mspace{14mu} {{{where} - \left( {c_{x};c_{y}} \right)} = {\frac{4}{1/4^{2}}\text{?}{K_{c_{x}}(x)}{J_{c_{y}}(y)}{f\left( {x;y} \right)}\frac{d_{x}d_{y}}{{h_{x}^{0}(x)}{h_{y}^{0}(y)}}\text{:}}}$?indicates text missing or illegible when filed

The scan frequency equation for 2DKT and the scan sensor output aregiven by:The scan frequency equation for general Lissajous scan is given by:

$f_{s} = {\frac{1}{T}\left( {{c_{x}n_{x}} + {c_{y}n_{y}}} \right)}$

The scan number equation for the general Lissajous scan is given by:

n _(s) =c _(x) n _(x) +c _(y) n _(y).

For example, in one embodiment, h_(x) and h_(y) are provided to be:

${15\; {h_{x}(t)}} = {{h_{y}(t)} = {{h(t)} = {\text{?}\begin{matrix}{{{\text{?} + 1} = 2},{0 \cdot t \cdot r}} \\{{{\text{?} + 1} = 2},{{r \cdot t \cdot 1_{i}}r}} \\{{{\text{?} + 1} = 2},{1_{i}{r \cdot t \cdot 1}}}\end{matrix}}}}$ ?indicates text missing or illegible when filed

FIG. 10 shows an example of a Lissajous-like scan path, in which h(t)two arcs are connected at the two ends of the strait line to keep thefunction smooth. The scan path is shown in FIG. 10. One advantage ofthis scan path is that the majority of the center part is evenlydistributed straight lines which enable an evenly distributed datacollection. At the same time, it maintains both the scan path and thescan velocity smooth which makes scan motion easier.

Daisy Scan

In one embodiment, where the target is round, a scan path may beprovided as

The associated spatial transform is given by:

g(r;μ)=″G(f _(r) ;f _(μ))e ^(if) ^(μ) ^(μ) T _(f) _(r) (r)

The scan frequency equation is given by:

$f_{s} = {\frac{1}{T}\left( {{f_{\mu}n_{\mu}} + {f_{r}n_{r}}} \right)}$

The scan number equation is given by:

n _(s) =f _(μ) n _(μ) +f _(r) n _(r)

FIG. 11 illustrates an example of a daisy scan path.

Symmetric Properties of Spatial Transform

For a Ping-pang Scan, a Lissajous Scan (or another Lissajous-like scan)and similar scan paths, the associated transforms are symmetrical. Forexample for 2DCT,

C(f _(x) ;f _(y))=C(_(i) f _(x) ;f _(y))=C(f _(x);_(i) f _(y))=C(_(i) f_(x);_(i) f _(y))

Therefore, when one of the four values are known, all four values areknown. For a Torus Scan, the 2DFT of the target has conjugate symmetry:

g(f _(x) ;f _(y))=conj(f _(t);_(i) f _(y))

g(f _(x) ;f _(y))=conj(_(i) f _(x) ;f _(y))

These symmetry properties may be used to calculate the spatial frequencycomponents from the 1DFT of the sensor output.

Calculating Two Spatial Components from One 1DFT Components of theSensor Output

For a Pingpang Scan, a Lissajous Scan (or a Lissajous-like scan) andsimilar scan paths, the spatial transform has real value. The 1DFT ofthe scan path is a complex value (i.e., having both a real part and animaginary part). Hence, two spatial components that contribute to thesame 1DFT component can be reconstructed from solving a set of 2 linearequations. One way for calculating the spatial components is to find theoffset of the scan path x₀; y₀ so that the conditional number of thelinear equations is small.

For example, when y₀=0, the linear equation is.

$\begin{matrix}{< {G_{s}\left( f_{s} \right)}^{\prime}} \\{= {G_{s}\left( f_{s} \right)}}\end{matrix} = {4\; Q\begin{matrix}{C\left( {f_{x\; 1};f_{y\; 1}} \right)}^{\prime} \\{C\left( {f_{x\; 2};f_{y\; 2}} \right)}\end{matrix}\mspace{14mu} {where}}$ $Q = {\begin{matrix}{< {q\left( f_{x\; 1} \right)}} & {< {q\left( f_{x\; 2} \right)}^{\prime}} \\{= {q\left( f_{x\; 1} \right)}} & {= {q\left( f_{x\; 2} \right)}}\end{matrix}\mspace{14mu} {where}}$${q\left( f_{x} \right)} = ^{{{1}/4}\; f_{x}\frac{x_{0}}{x}}$

The offset to for optimal conditional number is given by:

When

$\frac{n_{y}x_{0}}{x} = {\frac{1/4}{2}.}$

The Scan Number Equation Design Scan Numbers

In one embodiment, (n_(x); n_(y)) is provided so that there is only one2DFT frequency component that contributes to one 1DFT frequencycomponent at n_(s). One way to provide such (n_(x); n_(y)) is to find(n_(x); n_(y)) that satisfies:

f _(x) n _(yi) f _(y) n _(x) <F _(T) +f _(x) n _(yi) f _(y) n _(x) >F_(Ti) F _(T+i) F _(T) _(i) ·(n _(x) ² +n _(y) ²)  (1)

Since F_(T+) and F_(T) _(i) grow linearly with n_(x) and n_(y), and theright-hand side of the third inequality grows quadruply with n_(x) andn_(y), if f_(x) and f_(Y) are bounded (i.e., when the amplitude of the2DFT is significant only when F_(x) _(i) ·f_(x)·F_(x+) and F_(y) _(i)·f_(y)·F_(y+)), one can always increase n_(x) and n_(y) until all threeinequalities are satisfied.

In one embodiment, for all significant 2DFT terms, f_(x) and f_(y) arelimited by:

${\frac{{}_{}^{}{fx}_{}^{}}{N_{y}} + \frac{{}_{}^{}{fy}_{}^{}}{N_{x}}} < {1\text{:}(2)}$

One can then find any two relatively prime numbers n_(x) and n_(y) thatsatisfy

n_(x),N_(x)

n_(y),N_(y)

Since, from equation (2) one obtains F_(T+i) F_(T) _(i) ·(n_(x) ²+n_(y)²).

In one embodiment, the 2DFT terms are band-limited (i.e., significantonly at F_(X)+²>f_(x)>²)

where F_(X) is the bandwidth.

From (1),

$F_{T +} = {\left( {F_{x} + \text{?}} \right)\frac{n_{x}^{2} + n_{y}^{2}}{n_{y}}i\mspace{14mu} \left( {{f_{x}n_{x}} + {f_{y}n_{y}}} \right)\frac{n_{x}}{n_{y}}}$$\begin{matrix}{F_{T_{i}} = {\frac{{2n_{x}^{2}} + n_{y}^{2}}{n_{y}}i}} & {\left( {{f_{x}n_{x}} + {f_{y}n_{y}}} \right)\frac{n_{x}}{n_{y}}}\end{matrix}$ ?indicates text missing or illegible when filed

Hence,

${F_{T +}i\; F_{T_{i}}} = {\frac{F_{x}}{n_{y}}\left( {n_{x}^{2} + n_{y}^{2}} \right)}$

Therefore, when one has n_(y), F_(x), only one 2DFT frequency componentscan contribute to each 1DFT frequency component of the scan output.

In one embodiment, the 2DFT terms are band-limited in both the x and ydirections:

F _(X)+² _(X) >f _(X)>² _(X);

F _(Y)+² _(y) >f _(x)>² _(y);

Thus, from the previous example, only one 2DFT frequency component cancontribute to each 1DFT frequency component of the scan output, when:n_(y), F_(X) or n_(x), F_(Y). For example, when F_(X)=128 and F_(Y)=128,for all the following (n_(x); n_(y)) pairs, there is only one 2DFTcomponent that contribute to one 1DFT component of the scan output:(128, 127), (128, 95), (128, 63), (128, 31), (128, 1), and (127, 128),(95, 128), (63, 128), (31, 128), (1, 128).

Solving the Scan Number Equation

The scan number equation can be solved by finding (f_(x); f_(y)), or aset of (f_(x); f_(y)) for a given n_(x), n_(y), and n_(s) is known fromNumber Theory. One way is to build a look-up table from all possible(f_(x); f_(y)) to n_(s) for each, (n_(x); n_(y)) so that thecorresponding (f_(x); f_(y)) may be looked up from n_(s).

Calculating the Target from the Scan Sensor Output

Data collected from scan sensors may be sent to a data processing systemto recover the original target.

The Sparse Sampling Recovery Method

In one embodiment, the target is assumed to be sparse. For example, inone embodiment, the target is assumed sparse in a certain transformedspace of the target (“the sparse space”; i.e., a significant amount ofdata points are zeros, or close to zeros, or just noise). In anotherexample, in one embodiment, the target is assumed have a limitedvariance.

Sparse sampling recovery techniques may be used to recover the targetfrom the scan sensor output. Many sparse sampling recovery methods areknown to those skilled in the art. For example, “11-magic” is a softwarepackage suitable for solving sparse sampling recovery problems.“11-magic” may be obtained from at http://www.acm.caltech.edu//11magic/or from its creators, E. Cands and J. Romberg. One example of a sparsetarget to which the sparse sampling recovery method is applicable is thesky in dark night: most of the target is black except for the luminousstars. In this case, the target can be recovered by solving the convexproblem

min kzk_(i) subject to kAz_(i)bk₂·²

where k¢k₁ denotes the I₁ norm. k¢k₂ and denotes the I₂ norm,

Represents the allowed error, z is the target, b is the measurementresult, and A is the operator that maps the target to the measurement.

As another example, the target is sparse in the wavelet transformedspace: i.e., the image's wavelet transform components contain many termsthat are zero or close to zero. In that case, the image can bereconstrcuted by solving the convex problem:

min kWzk₁ subject to kAz_(i)bk₂·²

where W denotes the wavelet transform operator. For example, when thetarget is mainly continuous except for some cutting boundaries, arecovery technique of total variance can be used. The original targetcan be calculated by solving:

min kzk_(TV) subject to kAz_(i)bk₂·²

where

is the total variance operator.

Adjusting the Error Level

The error level

may be determined using the noise level of the sensor.

An Example Using Ping-Ping Scan Paths

In one embodiment, the image is sparse in the wavelet transform space.In that case, multiple Ping-ping scan paths may be used. An operator maycalculate the 1DFT components of the sensor output from the target usingthe scan number equation; the b 1DFT scan sensor output may becalculated by FFT. The target may then be calculated by solving minkWzk₁ subject to kAz_(i) bk₂·² where

can be determined by pre-known sensor noise, A is the DCT transformoperator with component selection (i.e., only selecting the terms thathas an item in b).

An Example Using a General Scan Path

In one embodiment, the scan path may be a random-walk on the target,using a low-pass sensor. In this case, the sensor may be modeled as alinear time-invariant system whose output is a linear function oftarget. The operator A then becomes an operator that maps the target tothe sensor output. Vector b becomes the sensor output time history. Ifthe target is assumed sparse in a spatial domain, min kzk₁ subject tokAz_(i) bk₂·² can be used to calculate the target. In general, one canmove the scan sensor in any scan path and apply a similar method tomeasure the target. However, such a method may not be desirable because,first, the data collection is not efficient and, second, the requirementfor calculating the A operator is high.

System Operations

The system may operated according to the following three steps.

-   -   1. Scanning the target by moving the sensing location on the        target and measuring the sensor output and the sensing location        (The sensing location may be moved at a higher speed than        required by the Nyquist theorem);    -   2. Based on the measurement mechanism of the scan sensor,        finding a relationship between the target and the scan output;        and    -   3. Reconstructing the original target based on this relationship        and the scan sensor output.

In one embodiment, the system may be operated according to thefollowing:

-   -   1. scanning the target using multiple scan paths:        -   Based on different scanning and sensing machines, one may            select from many frequency mapping scan paths (e.g., a torus            scan, a ping-pang scan, a Lissajous scan, a Lissajous-like            scan, or a daisy scan). In one embodiment, the scan numbers            n_(x) and n_(y) are chosen such that each 1DFT term of the            scan sensor output is contributed by one (or a finite number            of) spatial frequency component of the target. The scan            numbers may be selected using the method discussed in            section Design Scan Numbers.    -   2. Choosing a scan speed;        -   The scan speed is chosen to adapt to the number of spatial            components that are being measured in the sensing band of            the scan sensor. If the scan speed is faster, the number of            spatial components measured is smaller.    -   3. scanning the target along all the different scan paths and        collecting the scan output;        -   During each scan, the data is collected after the sensor            reaches a steady state.    -   4. Taking the Fourier transform of the scan output;        -   Since the sensor output is periodic, the FFT of the sensor            output may be taken for a duration that is an integer            multiple of the scan period. The scan sensor may be            calibrated, its dynamic response is measured and the            sensor's output is normalized in frequency domain. If the            sensor's frequency response is S(f) and the FFT of the raw            sensor output is Sr(f), the normalized scan output is

${S_{n}(f)} = {\frac{S_{r}(f)}{S(f)}.}$

-   -   5. Calculating the spatial domain components from 1DFT of the        scan output using scan number equation; and    -   6. Reconstructing the target using a sparse sampling recovery        algorithm from the sparsely sampling spatial components.

Design Scan Numbers

Scan numbers may be selected by the following method:

-   -   1. estimating the highest spatial frequency of the target,        denote as F_(x), F_(y);    -   2. finding a set of directions along which the spatial        components of the target may be measured by each scan path; and        -   For example, in one embodiment, when the target is roughly            symmetric, the project directions are provided as evenly            distributed lines. From the scan frequency equation, the            directions of the strips are along directions with slope in            space.

$k = {i\frac{n_{x}}{n_{y}}\mspace{14mu} \left( {f_{x};f_{y}} \right)}$

-   -   3. for each project direction, denoting its slope as k,        selecting n_(x) and n_(y) so that

$i\frac{n_{x}}{n_{y}}\frac{1}{4}k$

-   -   and    -   n_(x); n_(y) are relatively prime, and    -   n_(x), F_(y) or n_(y), F_(x).    -   Such n_(x); n_(y) may be constructed by the following method:    -   1. if

$k > {j\frac{F_{x}}{F_{y}}j\text{:}}$

-   -   -   a. choosing n_(x) as the smallest integer that is the a            power of 2 and x, F_(y) (e.g., if F_(y)=101, choose            n_(x)=128:,); and        -   b. choosing n_(y) as the odd integer that is closest to            kn_(x) (e.g., if k=1, and n_(x)=128 choose n_(y)=127); and

    -   2, if

$k < {j\frac{F_{x}}{F_{y}}j\text{:}}$

-   -   -   a. choosing n_(y) as the smallest integer that is the a            power of 2; and n_(y), F_(x); and        -   b. choosing n_(x) as the odd integer that is closest to            kn_(y).

Iterative Operation

In one embodiment, the number of scan paths is determined in aniterative way:

-   -   1. selecting a few scan paths and calculating the target;    -   2. If the target is calculated successfully then stop.    -   3. Otherwise, adding another one or more different scan lines        and calculating the target again;    -   4. Repeating steps 1-3 until the target is successfully        calculated.    -   5. once the number scan paths is obtained for one target, that        same number of scan paths may be used as an initial value for        the next target.

The initial number of scan paths may also be decreased.

In one embodiment, the scan speed may be determined in an iterative way:

-   -   1. selecting an initial scan speed and calculating the target;    -   2. if the target is calculated successfully then stop.    -   3. Otherwise, decreasing the scan speed and calculate the target        again.    -   4. Repeating steps 1-3 until the target is successfully        calculated.    -   5. Once the scan speed is obtained for one target, the same scan        speed can be used as the initial value for the next target.

The initial number of scan speeds can also be increased.

Scan System with Multiple Scan Sensors

In some embodiments, more than one scan sensors may be used in thesystem. In one embodiment, a linear array of CCD sensors is provided fordocument scanning. In another embodiment, a two-dimensional CCD imagechip can be used as the scan sensors. In this system, the scan movementis done either by moving the image chip, or by moving the image, forexample, using a moving mirror. In one embodiment, the scan path of eachscan sensors is identical except its offset. Such an arrangement makesthe scan motion easier to implement.

In one embodiment, the offsets of the scan paths of different sensorsare selected so that their scan paths do not overlap. In anotherembodiment, the scan paths are evenly distributed on the target.

In one embodiment, for each sensor, multiple spatial frequencycomponents contribute to each 1DFT component. In one embodiment, thenumber of spatial frequency components contributing to each 1DFTcomponent equals the number of scan sensors multiplied by the number ofoptimal spatial components that contribute to each 1DFT components for asingle sensor scan system.

In one embodiment, multiple scan rounds may be provided. In each scanround, the sensors follow a scan path together, but with differentoffsets. Different scan paths for different rounds may be provided insubstantially the same way as a systems with one scan sensor.

During data processing, the measurement operator A maps the target todifferent 1DFT components from different sensor output in different scanrounds. For each sensor, the scan number equation is used to calculatethe 1DFT components in A, where b is the vector of 1DFT components fromdifferent sensor outputs in different scan rounds. When calculating thespatial frequency components for each sensor, only the area of thetarget that is scanned by that sensor is required.

The above detailed description is provided to illustrate the specificembodiments of the present invention and is not intended to be limiting.Numerous modifications and variations within the scope of the inventionare possible. The present invention is set forth in the accompanyingclaims.

1. A scan system for scanning a target, comprising: a scan sensorproviding scan output; mechanism for moving a sensing location of thescan sensor on the scan target; and a data processing system thatcalculates a physical parameter of the scan target from the scan output,wherein a bandwidth of the scan output is higher than a bandwidth of asensing band of the scan sensor.
 2. A scan system as in claim 1, whereinthe scan sensor comprises a linear array of sensor elements.
 3. A scansystem as in claim 1, wherein the scan sensor comprises a 2-dimensionalarray of sensor elements.
 4. A scan system as in claim 1, wherein themechanism moves the sensing location over multiple scan paths.
 5. A scansystem as in claim 4, wherein the scan paths comprise one or mreo closedscan paths.
 6. A scan system as in claim 4, wherein the scan pathscomprise one or more continuous scan paths.
 7. A scan system as in claim4, wherein frequency components of the sensor output are related by alinear relation to a spatial transform of the target.
 8. A scan systemas in claim 7, wherein spatial frequency components of the sensor outputare selected according to the linear relation.
 9. A scan system as inclaim 7, wherein the linear relation is used in fast data processing bythe data processing system
 10. A scan system as in claim 7, wherein atleast one of the scan paths has a portion of its spatial frequencycomponents contributing to the frequency components in the scan sensor'ssensing band.
 11. A scan system as in claim 10, wherein the scan pathscomprise one or more linear selecting scan paths.
 12. A scan system asin claim 10, wherein the scan paths comprise one or more projecting scanpaths.
 13. A scan system as in claim 12, wherein the scan paths includeone or more Torus scan paths.
 14. A scan system as in claim 13, whereinthe spatial transform comprises a Fourier transform.
 15. A scan systemas in claim 12, wherein the scan paths comprise one or mreo Ping-pongscan paths.
 16. A scan system as in claim 15, wherein the spatialtransform is a Cosine Transform.
 17. A scan system as in claim 12,wherein the scan paths comprise one or more Lissajous scan paths.
 18. Ascan system as in claim 17, wherein the spatial transform comprises aFast Fourier Transform.
 19. A scan system as in claim 12, wherein thescan paths comprise one or more Lissajous-like scan paths.
 20. A scansystem as in claim 19, wherein the spatial transform is a Fast FourierTransform.
 21. A scan system as in claim 12, wherein the scan pathscomprise one or more Daisy Scan paths.
 22. A scan system as in claim 21,wherein the scan paths include a finite number of spatial frequencycomponents that contribute to each frequency component of the scansensor's output.
 23. A scan system as in claim 22, wherein that finitenumber is 1
 24. A scan system as in claim 22, wherein that finite numberis
 2. 25. A scan system as in claim 24, wherein each component of thespatial transform has a real value
 26. A scan system as in claim 25,wherein the two real components are derived from a single frequencycomponent of the scan sensor output
 27. A scan system as in claim 26,wherein a scan path offset is adjusted so that linear equations usedhave an optimal conditional number
 28. A scan system as in claim 22,wherein the scan system comprises more than one sensor, and wherein thenumber of spatial frequency components contributing to each frequencycomponent of the scan sensor's output is more than one.
 29. A scansystem as in claim 28, wherein the number of spatial frequencycomponents that contribute to each frequency component of the sensoroutput is proportional to number of scan sensor used.
 30. A scan systemas in claim 28, wherein the number of spatial frequency components thatcontribute to each frequency component of the sensor output equal tonumber of scan sensors.
 31. A scan system as in claim 1, wherein aFourier transform of the sensor output is calculated in the dataprocessing system.
 32. A scan system as in claim 31, wherein a Fouriertransform is calculated for a full cycle of a closed scan path.
 33. Ascan system as in claim 1, wherein a dynamic response of the scan sensoris normalized by the data processing system.
 34. A scan system as inclaim 1, wherein the physical parameter of the target is calculatedusing a sparseness of the target.
 35. A scan system as in claim 33,wherein a linear relation exists between the target and the scan sensoroutput.
 36. A scan system as in claim 34, wherein the target is sparsein a transformed space.
 37. A scan system as in claim 34, wherein thephysical parameter of the target is calculated using a scan numberequation.